scholarly journals Modeling of heat and mass transfer processes in non-linearly viscous fluid film flows

2019 ◽  
Author(s):  
I. S. Tonkoshkur
Keyword(s):  
Mass Transfer ◽  
Differential Equations ◽  
Viscous Fluid ◽  
Film Thickness ◽  
Liquid Film ◽  
Heat And Mass Transfer ◽  
Boundary Layers ◽  
The Body ◽  
System Of Differential Equations ◽  
And Diffusion

The problem of heat and mass transfer in a liquid film of a nonlinearly viscous fluid flowing down the surface of a body of revolution under the influence of gravity is considered. The axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. It is assumed that the thermal and diffusion Prandtl numbers are large and the main changes in the temperature and diffusion fields occur in thin boundary layers near the solid wall and near the free surface separating the liquid and gas. A curvilinear orthogonal coordinate system (ξ, η, ζ) connected with the surface of the body is introduced. To describe the flow of a liquid film, a model of a viscous incompressible liquid is used, which is based on differential equations in partial derivatives - the equations of motion and continuity. As boundary conditions, the conditions of adhesion are used on the surface of a solid body, as well as the conditions of continuity of stresses and the normal component of the velocity vector - on the surface separating the liquid and gas. To simulate heat and mass transfer in a liquid film, the equations of thermal and diffusion boundary layers with boundary conditions of the first and second kind are used. To close the system of differential equations, the Ostwald-de-Ville rheological model is used. To simplify the system of differential equations, the small parameter method is used, in which the relative film thickness is selected. It is assumed that the generalized Reynolds number is of the order of unity. The solution of the equations of continuity and motion (taking into account the main terms of the expansion) is obtained in an analytical form. To determine the unknown film thickness, an initial-boundary-value problem is formulated for a first-order partial differential equation. The solution to this problem is found numerically using a running count difference scheme. To reduce the dimension of the problem for the equations of the boundary layer, the local similarity method is used. To integrate simplified equations, the finite-difference method is used.

scholarly journals Thermodynamics of non-isothermal diffusion at the extraction from cheese whey by Lupin

2019 ◽  
Vol 81 (3) ◽  
pp. 39-42
Author(s):  
Y. I. Shishackij ◽  
A. S. Belozercev ◽  
A. M. Barbashin ◽  
S. A. Nikel
Keyword(s):  
Mass Transfer ◽  
Temperature Gradient ◽  
Differential Equations ◽  
Heat And Mass Transfer ◽  
Thermal Diffusion ◽  
Cheese Whey ◽  
Driving Forces ◽  
Extraction Process ◽  
The Body ◽  
Shape Sphere

In many cases, extraction is accompanied by thermal phenomena. We have established the possibility of intensifying the process through the use of heated cheese whey. Lupine has a geometric shape (sphere, cylinder, plate) loaded into an extractor filled with cheese whey. Due to the temperature difference between the solid and the liquid, temperature gradients are observed. As the body warms up, the temperature gradient decreases and then disappears. For example, an organized step temperature mode. However, such a regime should be technologically and energetically justified. Thus, during extraction there is a periodic non-stationarity. The emergence of this period is noted in the main works. The expression for the increase in entropy per unit time is written. Given the changes in entropy, the Gibbs equation is written. The basics of equations are the second laws of thermodynamics. As a result, the results obtained as a result of thermodynamic driving forces were obtained. The equations of energy (heat) and mass transfer of substances are written. Thermodynamic forces contribute to the formation of heat flux and mass flow of substances. The consumption of a substance depends not only on the gradient (diffusion), but also on the temperature gradient (thermal diffusion). Air temperature is defined as a temperature gradient. The differential equations of heat and mass transfer of Lykov were rewritten taking into account the extraction process. The numerical values of the coefficients Dт and aс they relate to the assessment of the effect of superposition effects (thermal diffusion and diffusion thermal conductivity). The overlay effect can be neglected, since the relatively small gradients of temperatures and concentrations arising in the lupine. It is noted that the possibility of simplified differential equations is associated with small values of the Lykov criterion. Because of this, there should be little.

scholarly journals Mathematical modeling of film flow of a liquid on a surface of a body of a rotation

2018 ◽  
Author(s):  
I. S. Tonkoshkur
Keyword(s):  
Differential Equations ◽  
Film Thickness ◽  
Small Parameter ◽  
Rheological Model ◽  
The Body ◽  
Viscoplastic Fluid ◽  
Parameter Method ◽  
Order Partial Differential Equation ◽  
System Of Differential Equations ◽  
Partial Differential

The problem of the spatial nonwave stationary flow of the viscoplastic fluid on the surface of the body of rotation under the action of gravity is considered. It is assumed that the axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. A curvilinear orthogonal coordinate system (ξ, η, ζ) associated with the body surface is introduced: ξ is the coordinate along the generatrix of the body, η is the polar angle in the plane perpendicular to the axis of the body of revolution, ζ is the dis-tance along the normal to the surface. To describe the flow of a liquid film, a viscous in-compressible fluid model is used, which is based on partial differential equations - the equations of motion and continuity. The following boundary conditions are used: sticking conditions on the solid surface; on the surface separating liquid and gas, the conditions for continuity of stresses and normal component of the velocity vector. For the closure of a system of differential equations, the Schulman rheological model is used, which is a gener-alization of the Ostwald-de-Ville power model and the Shvedov-Bingham viscoplastic model. To simplify the system of differential equations, the small parameter method is used. The small parameter is the relative film thickness. It is assumed that the generalized Reynolds number has an order equal to one. The solution of the equations of continuity and motion (taking into account the principal terms of the expansion) was obtained in an analytical form. The obtained formulas for the components of the velocity and pressure vector generalize the known relations for flat surfaces. To determine the unknown film thickness, an initial-boundary value problem was formulated for a first-order partial differential equation. The solution to this problem is found with the help of the finite difference method. The results of calculations according to the proposed method for the circular cone located at a certain angle to the vertical are presented. Calculations show that the parameters of nonlinearity and plasticity of this rheological model of a liquid can significantly affect the speed profiles and the distribution of the thickness of the viscous layer on the surface of the body

scholarly journals Second Law Analysis of a Gas-Liquid Absorption Film

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Nejib Hidouri ◽  
Imen Chermiti ◽  
Ammar Ben Brahim
Keyword(s):  
Mass Transfer ◽  
Film Thickness ◽  
Liquid Film ◽  
Heat And Mass Transfer ◽  
Entropy Generation ◽  
Gas Absorption ◽  
Heat Transfer Fluid ◽  
Second Law ◽  
Coupling Effects ◽  
Liquid Absorption

This paper reports an analytical study of the second law in the case of gas absorption into a laminar falling viscous incompressible liquid film. Velocity, temperature, and concentration profiles are determined and used for the entropy generation calculation. Irreversibilities due to heat transfer, fluid friction, and coupling effects between heat and mass transfer are derived. The obtained results show that entropy generation is mainly due to coupling effects between heat and mass transfer near the gas-liquid interface. Total irreversibility is minimum at the diffusion film thickness. On approaching the liquid film thickness, entropy generation is mainly due to viscous irreversibility.

scholarly journals Modeling the gravitational flows of a viscoplastic fluid on a conical surface

2019 ◽  
Author(s):  
I. S. Tonkoshkur ◽  
K. V. Kalinichenko
Keyword(s):  
Differential Equations ◽  
Film Thickness ◽  
Small Parameter ◽  
Cross Section ◽  
The Body ◽  
Viscoplastic Fluid ◽  
Parameter Method ◽  
Order Partial Differential Equation ◽  
System Of Differential Equations ◽  
Partial Differential

рідка плівка,The problem of a stationary waveless gravitational flow of a viscoplastic fluid over the surface of a cone with an arbitrary smooth cross section is considered. It is assumed that the axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. A curvilinear orthogonal coordinate system (ξ, η, ζ) associated with the body surface is introduced: ξ is the coordinate along the generatrix of the body, η is the polar angle in the plane perpendicular to the axis of the body of revolution, ζ is the distance along the normal to the surface. To describe the flow of a liquid film, a viscous incompressible fluid model is used, which is based on partial differential equations - the equations of motion and continuity. The following boundary conditions are used: sticking conditions on the solid surface; on the surface separating liquid and gas, the conditions for continuity of stresses and normal component of the velocity vector. To close the system of differential equations, the Shvedov-Bingham rheological model is used. To simplify the system of differential equations, the small parameter method is used. The small parameter is the relative film thickness. It is assumed that the generalized Reynolds number has an order equal to one. The solution of the equations of continuity and motion (taking into account the principal terms of the expansion) was obtained in an analytical form. The obtained formulas for the components of the velocity and pressure vector generalize the known relations for flat surfaces. To determine the unknown film thickness, an initial-boundary value problem was formulated for a first-order partial differential equation. The solution to this problem is found with the help of the finite difference method. The results of calculations by the proposed method for cones with a cross section in the form of a circle and a square with rounded corners are presented. Calculations show that the plasticity parameter and the cross-sectional shape significantly affect the velocity and distribution profiles of the thickness of the viscous layer over the surface of the body.

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